The localization of 1-cohomology of transitive Lie algebroids (Q854655)

For a transitive Lie algebroid \(A\) on a connected manifold \(M\) and its representation on a vector bundle \(F\), a morphism of cohomology groups \(\gamma^k: H^k(A,F)\to H^k(L_x, F_x)\), the localization map, is defined. \(L_x\) is the adjoint algebra at \(x\in M\). The main result is the following: if \(M\) is simply connected, or \(H^0(L_x,F_x)\) is trivial, then \(\gamma^1\) is injective and the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases. See also \textit{A. Connes} [J. Math. Phys. 36, No. 11, 6194--6231 (1995; Zbl 0871.58008)] and \textit{A. Weinstein} [Lett. Math. Phys. 56, No. 1, 17--30 (2001; Zbl 1010.53060)].